Flocking dynamics with voter-like interactions: speed-dependent transition and fast polar consensus

Flocking dynamics with voter-like interactions: speed-dependent transition and fast polar consensus

Abstract

We study the collective motion of self-propelled particles with voter-like interactions. Each particle moves at a constant speed on a two-dimensional space and, in a single step of the dynamics, it aligns its direction of motion with that of a randomly chosen neighboring particle. Directions are also perturbed by an external noise of amplitude . We find that the global alignment of particles, measured by the order parameter , depends crucially on the particles’ speed. At low speeds, any amount of noise keeps the system totally disordered () in the thermodynamic limit, while full polar order () is achieved for zero noise . At high speeds, a very different ordering behavior is found, where various observables reveal an order-disorder phase transition of second order at a finite critical noise . We also investigate the evolution to complete order (polar consensus) at zero noise. We find that increases as for short times and approaches exponentially fast to for long times. The mean time to consensus is non-monotonic with the density of particles, and for medium and high densities the consensus is faster than in the case of all-to-all interactions. We show that the fast consensus is a consequence of the segregation of the system into clusters of equally-oriented particles, which breaks the balance of transitions between directional states observed in well mixed systems.

1 Introduction

The study of the collective properties of systems composed by self-propelled individuals has been focus of intense research for the last two decades [1, 2, 3]. The flocking behavior of a large group of animals is observed in many different species such as birds, fish, bacteria and insects. Among the theoretical studies, the model proposed by Vicsek [4] is one of the simplest physical models of self-propelled interacting particles that is able to reproduce the collective behavior observed in real biological systems. The interaction rules of most flocking models are based on the standard Vicsek model (SVM), in which particles tend to align their direction of motion with the average direction over its local neighborhood. That is, each particle interacts with a group of particles that could be large or small, depending on the local density of particles and the specific rules of the model. The size of the interacting group plays an important role in the dynamics of decision making in human or animal populations [5, 6]. For instance, it was shown in [7] that small groups of individuals can maximize the accuracy of decisions.

In this work we explore the simplest case scenario of pairwise interactions. We propose and study an alternative to the SVM where, in a single iteration step, each particle tries to align its direction of motion with that of only one particle at a time, chosen at random within its interacting neighborhood, and in the presence of a small perturbation. If we take the example of a flock of birds, this model can be interpreted as each bird adopting the direction of one close-by bird even if it is able to see a group of, lets say, birds around. This dynamics of velocity imitation is reminiscent of the voter model for opinion formation if one interprets animals (or particles) as individuals (or agents), and directions as opinions. In the original voter model (VM) [8, 9] each individual or voter is located at the site of square lattice, and can take one of two possible opinions. Voters’ opinions evolve by an imitation mechanism that consists on blindly adopting the opinion of a random nearest neighbor. The alignment with local opinions leads to the formation of same-opinion domains that slowly (logarithmically) growth in size [10], until one domain takes over the entire lattice and the system reaches consensus in a time that scales as in two dimensions [11]. This ultimate state is frozen as opinions cannot longer evolve. This macroscopic dynamic behavior of the VM can be understood in terms of its associated Langevin equation for the magnetization field [12, 13], whose Ginzburg-Landau potential is zero. The VM with an arbitrary number of opinion states has recently been studied in [14, 15], where the authors found interesting properties on the evolution of the number of different opinions and the consensus times. Some recent works have also applied social interactions to study flocking, like the majority rule dynamics explored in [16], where interactions are not pairwise as in our model but they include a large group of neighbors as in the Vicsek Model.

For its part, the SVM can be thought as a non equilibrium (ballistic) version of the XY model, where particles move ballistically in the directions of their spins. In this respect, one of the most interesting results of the SVM is that it can sustain long-range order in two dimensions for finite values of noise due to the particles’ motion [17], unlike the XY model which cannot [18]. As it is known that adding thermal bulk noise in the VM destroys global order in any dimension [19] (even when the noise is very weak), the study of a ballistic version of the VM could help us to better understand the ordering properties of collective motion and its dependence on the type of interaction. We want to note as well that the flocking voter model (FVM) proposed here can be framed as a particular type of coevolving network model [20, 21, 22], in which the network of interactions evolve as particles enter and leave the interaction range of other particles.

In this article we show that the behavior of the flocking voter model is very different from that of the Vicsek model. First, the system is not able to sustain long-range order when particles move slow enough. As a consequence, at low speeds and for a broad range of densities, the FVM does not exhibit the order-disorder phase transition induced by noise observed in the SVM and its extensions. However, the system does exhibit an order-disorder phase transition of second order at high speeds. Second, the ordered phase seems to be of a different nature than that of the SVM, as the traveling bands found in the SVM and recently studied in [23] are not observed in the ordered phase of the present FVM. Third, at zero noise the ordering dynamics is much slower than that in the SVM. Also, the interplay between the interaction pattern and the alignment dynamics leads to a relaxation to the final ordered state that largely deviates from the one observed in mean-field, as well as in lattices with static particles. As a consequence, the mean time to reach full polar order (consensus) exhibits a non-monotonic dependence with the density of particles. The motion appears to accelerate the polar consensus by a non-trivial mechanism that breaks the equivalence between particle states typically observed in the voter dynamics, introducing a net drift from minority to majority states.

The article is organized as follows. We introduce the model and its dynamics in section 2. Then, in section 3 we study how noise affects order in the stationary state, at low and high speeds. In section 4 we explore the ordering dynamics for the zero-noise case. We analyze the consensus times to the final ordered state in section 5. Finally, we discuss the results and state the conclusions in section 6.

2 The Model

We consider a set of particles that move on a continuous two-dimensional space with periodic boundary conditions. The density of particles is conserved at all times. At a given time , the position of particle is denoted by and its velocity by , with speed and direction , for . That is, all particles move at the same speed but not necessarily in the same direction. At each time step of length , each particle selects a random neighboring particle inside a circular region of radius centered at , and updates its position and direction according to

(1a)
(1b)

where is the direction of the particle at time , and is a white noise uniformly distributed between , with the noise amplitude. The notation mod in Eq. (1b) denotes the congruent angle that lays in the interval , and is used to make sure that the updated angle does not fall off the bounds . In case particle has no neighbors inside the interaction range , then its direction is changed only by the noise. At , positions of particles are assigned randomly with a uniform distribution inside the box , while their directions are randomly chosen from the interval . Then, each particle moves at a constant speed following a given straight path and can update its direction at integer times , by adopting the direction of a random neighbor. We mention that we are choosing in our model the updating rule used in the original version of the Vicsek model [4], which is known under the name of backward update. In the backward update the velocity at time is used to obtain the position of a particle at the next time (Eq. 1a) , whereas in the forward update that position is obtained using the velocity at time . This ambiguity in the selection of the position update and their consequences in the dynamics of the Vicsek model were discussed in some works (see for instance [26]). However, we expect the qualitative behavior of the FVM to be the same under both updates at low speeds. Qualitative differences may appear at high speeds where the difference between moving a particle before or after adopting a direction could be very large in some cases.

In the next section we study the stationary state of the system under the dynamics described by Eqs. (1a) and (1b), and how it is influenced by the directional noise.

3 Ordering properties under the influence of noise

Voter-like interactions in the dynamics of the model tend to align the direction of neighboring particles. This leads to local order in the short run, and to global or macroscopic polar order in the long run, as it happens in flocking models with ferromagnetic interactions like the standard Vicsek model. The ordering properties in these systems are characterized by the parameter , defined as

(1b)

which is the absolute value of the normalized mean velocity of all particles. The order parameter can vary from (total disorder) to (full order). In the SVM, noise –in its various forms [4, 16, 24, 25]– plays a fundamental role in the behavior of the system. It is known that the amplitude of noise induces an order-disorder phase transition, from a phase where a large fraction of particles move in a similar direction (order) for small , to a phase in which particles move in random directions (disorder) for large . In order to explore this phenomenon in the FVM we run computational simulations of the dynamics for various speeds, densities and system sizes, and study the behavior of three different magnitudes with , which characterize the order of the system and its associated phase transitions. These magnitudes are the average order parameter at the stationary state , the susceptibility and the Binder cumulant , where is the th moment of . Averages were done over independent realizations. In the next two subsections we analyze separately the case of low and high speeds given that, as we shall see, the ordering properties of the system crucially depend on the speed of particles.

3.1 Absence of order-disorder phase transition at low speeds

In this subsection we analyze simulation results for speed and various densities. Figure 1 shows the behavior of , and in systems with speed and density . We observe in Fig. 1(a) that decays continuously with for all system sizes, and that decays to zero with for all values of (see inset). This shows that for a broad range of the order seems to vanish in the limit. Also, only at for all system sizes.

                          
                          
Figure 1: Finite-size scaling analysis of different magnitudes of the model for speed , density of particles and system sizes (circles), (squares), (diamonds), (up triangles) and (left triangles). (a) Average of the order parameter vs noise amplitude . Inset: scaling of with the system size in the limit, for a fixed value of and (from top to bottom). The straight dashed line has slope . (b) vs the rescaled noise on a double logarithmic scale showing the collapse of the data into a single universal curve. The solid line is the fitting function . Inset: vs on a double logarithmic scale. (c) Susceptibility vs . Inset: rescaled susceptibility vs rescaled noise on a double logarithmic scale showing the data collapse. (d) Binder cumulant vs on a log-linear scale. Inset: data collapse.

In order to explore whether this behavior strictly holds for any value of larger than zero we performed a finite-size scaling analysis. Given that the curves in Fig. 1(a) are shifted towards as increases, we rescaled the -axis with the system size by the factor , as we see in Fig. 1(b). As all curves collapse into a single universal curve that can be approximately fitted by the function , with , we can argue that the level of ordering as a function of the rescaled noise is independent on the system size. The scaling function corresponds to a reasonable fitting of the collapsed data, even though we are aware that better fittings functions may exist. Then, by increasing and decreasing such that their product remains fixed, we can keep the ordering in a fixed value. This means that for any constant the noise vanishes as , and thus curves for different sizes are shifted towards with [see inset of Fig. 1(b)], and they tend to a step function in the limit ( for and for ). The picture presented above is consistent with a phase transition at , as the scaling follows the mathematical form above the transition point , with .

The scaling function allows to obtain useful information on how vanishes in the limit. Indeed, decays as a power law for , and so for . Therefore, the scaling theory predicts that for any fixed the ordering vanishes with as in the thermodynamic limit. This power-law decay is plotted by a dashed line in the inset of Fig. 1(a), where we observe that matches quite well the decay of for large obtained from simulations for most values of . We can also check that complete order () is only obtained for ().

Figure 1(c) shows that the maximum value of the susceptibility (the peak) increases with the system size and that the location of the peak is shifted towards as increases. We can see in the inset that the shape of around the peak follows a universal curve for different system sizes when the -axis and -axis are rescaled by and , respectively. This is consistent with a transition at a noise value in the thermodynamic limit, as the peak of diverges as (with ) and its location vanishes as (with ). We note that the scaling exponent is in agreement with the one found for the scaling of [Figure 1(b)].

The behavior of the Binder cumulant in Fig. 1(d) is compatible with the absence of a phase transition at finite noise. That is, at the transition point all curves for different sizes either cross if the transition is of second order or exhibit a negative peak that increases with the system size if the transition is of first order. None of these behaviors are observed in Fig. 1(d). This observation is consistent with the conclusion obtained from the other magnitudes, that there is no transition at a finite value .

In order to study how the results reported above depend on the density we run additional simulations for speed and smaller and larger densities and , respectively. We found that the behavior of the different magnitudes , and are similar to the ones presented above for and , with the same scaling exponents (plots not shown here).

In summary, we conclude from simulations that for speed and densities in the range the system approaches to complete disorder in the limit for any value . The peak of the susceptibility shifts to as increases, suggesting that there is an order-disorder phase transition at zero noise (). This is also consistent with the behavior of the Binder cumulant, which does not show evidence of a transition at finite noise (). The same results showing the absence of a transition were also found for a larger speed (plots not shown). However, a different picture is observed when the speed increases to larger values, as we describe in the next subsection.

3.2 Order-disorder phase transition at high speeds

Here we discuss simulation results for a speed . Unlike it happens at low speeds ( analyzed in the last subsection), we shall see that when the speed is high the system undergoes an order-disorder phase transition at a finite noise, analogous to the one observed in the Vicsek model. In Fig. 2 we show results for , density and various system sizes. Figure 2(d) shows the behavior of the Binder cumulant vs . We observe that all curves cross at a given value , which is a clear signature of a second-order phase transition at . As we see in Fig. 2(a), the order parameter decays to zero with , and the curves for different system sizes can be collapsed into a single curve with the mathematical form [see Fig. 2(b)], where is the dimension of the system, is the critical noise, and and are critical exponents. Using the approximate value for obtained from Fig. 2(d), the best collapse of the data shown in Fig. 2(b) was found for the exponents and . Also, the behavior of the susceptibility shown in Fig. 2(c) is consistent with a second-order phase transition at , where the data is well collapsed using the exponents and , as we can see in the inset.

We have also run simulations for the same speed and densities and , and found in both cases a phase transition analogous to the one described above for , with very similar critical exponents (plots not shown). Thus, the critical behavior of the system for does not seem to be affected by the density. Simulation results for a smaller speed are qualitatively the same as the ones for , but with a smaller order-disoreder transition point (plots not shown).

                          
                          
Figure 2: Finite-size scaling analysis of different magnitudes of the model for speed , density of particles and system sizes (circles), (squares), (diamonds), (up triangles), (left triangles), (down triangles) and (right triangles). (a) Average of the order parameter vs noise amplitude . Inset: vs on log-log scale. (b) Collapse of the data from panel (a) (more details in the text). (c) Susceptibility vs . Inset: data collapse (details in the text). (d) Binder cumulant vs on a log-linear scale. Inset: zoom around the crossing point .

3.3 Remarks on the ordering with noise

We conclude this section by summarizing the main results about ordering for the values of speeds and densities reported above, which cover a broad range of parameters. At low speeds, the FVM does not exhibit the order-disorder phase transition induced by noise observed in the SVM and related models. The order goes to zero for any in the thermodynamic limit, and thus the system remains completely disordered. That is, in the absence of noise the system reaches full order, but a tiny amount of noise is enough to drive the system to complete disorder. This is perhaps related to the fact that any type of bulk noise in the voter model prevents global order [19]. It seems that the imitation dynamics of the FVM makes the ordered state very unstable under the presence of noise, and the slow motion of particles is not enough to reestablish the order. This is an important difference with the SVM where the motion is able to align a large fraction of particles and induce a transition to order. Simulations on fully-connected systems give results very similar to the ones for low speeds (not shown), but with scaling exponents and .

However, the ordering behavior is quite different when particles move faster. That is, at high speeds the FVM undergoes a transition between an ordered and a disordered phase. It seems that the fast motion of particles is able to align the directions of particles that are further apart, inducing a long-range order that leads to global order in the thermodynamic limit when the noise is low enough, as in the SVM.

From the analysis shown above, it is worth stressing that complete order is only achieved when noise is zero. In this case, the dynamics is akin to that of the multi-state voter model [14, 15] if we think directions as opinions, as each particle adopts the exact direction of a random neighbor. Here the system starts with many different opinions distributed over the particles and evolves until consensus of a single opinion is reached. Then, the typical magnitudes of interest are the evolution of the number of different opinions and the mean consensus time, which we study in the next sections.

4 Ordering dynamics at zero-noise

We are interested in studying the dynamics of ordering in the noise-free case scenario where, as we showed above, full order is eventually achieved in the long run. In this final ordered state all particles move in the same direction, thus no more direction updates are possible. In Fig. 3 we show the temporal behavior of the average order parameter for , , and two different densities. The two lower curves correspond to the FVM, while the two upper curves are for the SVM. We observe that the approach to complete order () is much slower in the FVM. This slow ordering in the FVM is characterized by an initial increase of as a power law in time, with an exponent close to (dashed line), and a final exponential approach to (full order). This is closely related to the behavior of the mean number of different directions at time shown in Fig. 4. Initially, all directions are different as they are randomly assigned, and thus . Then, for densities we see that decreases very slowly during an initial transient of order , as

(1c)

represented by a solid line in Fig. 4. This result was derived analytically in [14] following a Master Equation approach and in [15] using generating functions, for the multi-state voter model under sequential update on a complete graph (all-to-all interactions). In the synchronous version of the model that we use here time is rescaled by a factor . In the final regime relaxes exponentially fast to , corresponding to the single direction of full order (see Fig. 4). We note that the decrease of is monotonic at all times. This is because some directions may not be copied by any particle in a single step (mainly those directions followed by few particles), and thus these directions disappear from the system. Then, given that new directions are never created in the voter dynamics, just decreases monotonically with time.

Figure 3: (Color online) Time evolution of the average order parameter in a system with particles, with zero noise () and speed . Blue (upper) symbols correspond to results of the SVM for particle densities (triangles) and (diamonds). Red (lower) symbols are results for the FVM with densities (squares) and (circles). The solid line is the theoretical approximation Eq. (1k), while the dashed line is a guide to the eye with slope .

Solid lines in Fig. 4 correspond to Eq. (1c), which reproduces quite well the evolution of from simulations on a fully-connected system (all-to-all interactions), represented by empty circles. We also see that these mean-field (MF) curves agree very well with simulations on 2D for large particle densities. Indeed, as the box’s length decreases with , the MF limit of all-to-all interactions is achieved as , independent on , because approaches the interaction range . Deviations from MF are evident for very low values of (see curve), where the number of neighbors is very small and local interactions rule the dynamics. We also observe that for low speed [Fig. 4(a)] the agreement between MF and 2D is quite good for , but for high speeds [Fig. 4(b)] 2D results depart from the MF curve at a time around . We shall discuss this fast decay of for large speeds in section 5.

Figure 4: Mean number of different directions vs time at zero noise (), for a system of particles, speeds (a) and (b), and densities (circles), (squares), (diamonds) and (triangles). Empty circles correspond to numerical results of on a fully-connected system (), while solid lines are the analytical expression Eq. (1c). Insets: Plots of as function of time for densities and in panel (a), and , and in panel (b). The peaks of the curves make evident the departure of the FVM dynamics from the mean field case.

An approximate expression for the relation between and can be obtained by assuming that the directions of particles are randomly distributed at all times, as we describe below. Even though this assumption is only strictly valid at we shall see that it also works reasonable well for longer times. Let’s consider that, at a given time , there are different and independent directions in the system of particles, drawn uniformly in the range. The number of different directions at time may vary between realizations, but here we assume that they are all equal to its mean number . Then, we can rewrite the order parameter at time from Eq. (1b) in the form

(1d)

where the sum is over the different particles’ directions labeled by the index , which runs from to its total number . The number of particles moving in the direction is denoted by and is normalized at all times . As particles are initially assigned random directions, we have that and for all . The evolution of the occupation numbers is not trivial, thus we make an approximation and assume that at all times they are all equal to its mean value (). Therefore, Eq. (1d) can be written as

(1e)

where we have defined the resulting vector . Given the assumption that all velocities are independent and uniformly distributed in , from the central limit theorem we know that, in the limit, the average and the standard deviation of the component of are, respectively,

(1f)
(1g)

where we have used the expressions and for the average and standard deviation, respectively, of the component of , . Analogous expressions follows for the component of . Besides, we also know that should be normally distributed

(1h)

Now, we can calculate the average value of the order parameter by implementing polar coordinates, as

(1i)

where we have used . Finally, using we arrive to

(1j)

In Fig. 5 we plot vs obtained from computational simulations on a fully-connected system (empty circles) and on a 2D system, for and different densities, and compare with the behavior predicted by Eq. (1j) (solid line). At (S=N) the expression from Eq. (1j) works very well because all velocities are actually randomly distributed, and thus the occupation number distribution is uniform ( for all ). Then, as groups of particles start to have the same direction the distribution deviates from uniform, and thus the assumption for all implemented above does not hold any more. However, for short times –or large values of – the uniform approximation still works quite well, and the relation between and for 2D systems is well described by Eq. (1j), as we can see in Fig. 5.

Figure 5: Average number of different directions vs average order parameter at zero noise (), for a system of particles, speed , and densities (filled circles), (squares) and (triangles). Empty circles correspond to numerical results on a fully-connected system, while the solid line is the analytical estimation from Eq. (1j).

Combining Eqs. (1c) and (1j) we arrive to the following approximate expression for the time dependence of

(1k)

indicated by a solid line in Fig. 3. Even though there are discrepancies between the numerics and the analytical expression Eq. (1k) for , the algebraic increase for intermediate times (dashed line) predicted by Eq. (1k) seems to hold quite well for both particle densities. In Fig. 5 we can see that for a fixed value of , 2D simulations (filled symbols) show more orientational order than the corresponding MF simulation (empty circles), as is larger in the former case. This is a consequence of the fact that in 2D the different directions of particles at a given time are not uniformly distributed in the interval. Instead, longer lasting directions tend to be similar. This happens because orthogonal directions quickly tend to annihilate each other, given that interactions (and alignments) are more frequent between particles with high relative cross section. This can be seen in the frame of Fig. 8 where the two largest clusters move almost parallel to each other.

5 Consensus times

A magnitude of interest in models that exhibit a complete order is the time to reach the final ordered state or consensus time. In Fig. 6(a) we plot the mean consensus time over many realizations as a function of the particle density , for a system of particles and different speeds. In the static case scenario (circles) decreases with and approaches the MF value () for large [14, 15] (horizontal dashed line). As discussed in section 4, the MF limiting case is obtained when and thus each particle falls in the interaction range of any other particle. Therefore, for this happens when overcomes the value . However, already for is and the system behaves as in MF. In Fig. 6(b) we show the dependence of with the number of particles, for and different densities. For high densities, approaches the MF linear behavior (dashed line), but for low densities there are logarithmic deviations consistent with the 2D behavior [10, 11]. This crossover between 2D and MF can be better seen in the inset of Fig. 6(b), where we show vs on a log-linear scale to capture logarithmic corrections. We observe that data points fall on a straight line with density dependent -coordinate and slope . As increases, goes to zero and approaches , recovering the MF behavior.

We need to mention that while consensus is always achieved when particles move, some realizations never reach consensus when (specially at low densities), and thus those realizations were not considered in the calculation of . The reason of non-consensus states is because in the initial spatial configuration some individual particles and also clusters of particles might be isolated (they do not interact with the rest of the particles in the system). The lack of interactions throughout the system leads to a frustrated disordered state composed by more than one angular direction and, therefore, there is never consensus. A frustrated configuration can be either frozen (no more updates) or composed by isolated pairs of particles that interchange their directions ad infinitum (blinkers). Then, to calculate for we stopped each realization only when no more angular changes were registered, or when the system entered into a cycle of repeated configurations for long times (presence of blinkers). Then, we included in the calculation of only those realizations that did reach consensus, and disregard those realizations that did not.

                      
Figure 6: (a) Mean consensus time vs particle density at zero noise (), for a system of particles and speeds (circles), (squares), (diamonds) and (triangles). The horizontal dashed line denotes the mean field consensus time , while the upper dashed line is approximation from Eq. (1l). Inset: Same data on a linear-log scale showing the non-monotonic behavior regime. (b) vs for and densities (circles), (squares) and (diamonds). The dashed line is the expression . Inset: vs on a log-linear scale. Solid lines are the best linear fits , with coefficients and and slopes and for densities and , respectively.

The dynamic case scenario is different from the static case , as exhibits a non-monotonic behavior with . The mean consensus time is much larger than for low densities, but it decays to values smaller than as increases and, after reaching a minimum, increases and saturates at the MF value. As expected, becomes independent of at high densities. This peculiar shape is a consequence of the non-trivial interplay between particles’ speed and the propagation of directions over the space. We understand that the approach to consensus depends on the relation between the time scales associated to these two processes –convection and diffusion–, which vary with and . Below we explore in more detail the origin of the non-monotonic behavior of with . We first provide some scaling arguments that explain the behavior of with in the limit of small densities and large speeds, and then study the clustering dynamics at intermediate and high densities.

5.1 Limit of small densities and high speeds

Let us consider the case where particles move fast, and so they travel such a long distance in each time step that their distribution remains nearly uniform over the box. This is so because the backward update [26] used in simulations involves two consecutive events: each particle first aligns its direction with a neighboring particle and then moves a distance with the old direction. As a consequence, for large speeds aligned particles could be at any distance from each other, leading to a quite uniform spatial pattern. Also, we can assume that the system remains well mixed at high speeds and thus interactions are as in MF. Then, at each time step, direction updates take place only among those particles that have at least one neighboring particle inside their interaction range. For each particle this happens with probability for , assuming that particles are uniformly distributed. The fact that particles not always update their directions at every time step introduces a time delay in the alignment dynamics, which rescales the MF consensus time by a factor of (the mean number of attempts between interactions). Therefore, we arrive to

(1l)

We observe in Fig. 6(a) that the expression from Eq. (1l) (dashed line) gives a reasonable estimation of for high speed in the limit of low densities. The power-law decay at low is also observed for speed . However, Eq. (1l) overestimates the numerical data at intermediate densities, where the formation of clusters plays an important role, as we show in the next subsection.

5.2 Clustering dynamics at intermediate and high densities

For intermediate densities, the consensus turns to be faster than in MF. This is consistent with the behavior of the number of different directions in Fig. 4. There we can see that decays as in MF () during an initial transient, but then it starts to decay faster (exponentially), leading to a consensus time that is smaller than . This effect is more pronounced for large speeds [see Fig. 4(b)]. As we explain below, the departure observed in from a power law to an exponential decay is caused by a dynamic reordering of the spatial pattern of interactions, from a well mixed system to structures localized in space. To explore this in more detail, we studied the time evolution of the mean number of neighbors (mean degree). Results are shown in Fig. 7 for (the same as in Fig. 4), and various densities, where we observe that starts to increase very slowly from its initial value –corresponding to a uniform distribution of particles– until it saturates for long times when the system reaches consensus. This increase in the number of neighbors suggests that particles aggregate into spatial clusters. Indeed, when two particles are less than a distance apart they can align their directions and then move together until one of them changes direction by interacting with a third particle. Thus, one can see that alignment implies the “sticking” of nearby particles, forming large sets of particles that move together in the same direction. These structures can be seen in Fig. 8 for a system of particles, with and . The panels represent snapshots of the system at different times, showing the transition from a uniform distribution of particles moving in random directions at (a), to a spatial segregation into clusters of particles with the same direction at (b) and (c), and to a quasi-consensus state at . The segregation occurs for higher densities as well, but clusters may overlap when densities are very high.

Figure 7: Time evolution of the mean number of interacting neighbors at zero noise (), for , speed and densities (circles), (squares), (diamonds) and (triangles).
                 
                 
Figure 8: (Color online) Snapshots showing the configuration of the system at different times, composed by particles, with a density , speed and zero noise (). Particles are depicted by filled circles of radius , equal to 0.5, thus interacting particles overlap in this scale. Arrows indicate the direction of motion of each cluster, shown in a particular color.

By comparing Fig. 4(b) with Fig. 7, we observe that the deviation of from the MF value starts approximately when starts to be significantly larger than its initial value ; for instance at time for . This indicates that the formation of clusters speeds up the dynamics and the approach to consensus. It turns out that spatial segregation induces a drift in the transitions between directions, from directions followed by small clusters to directions of large clusters. In other words, large clusters are more likely to gain particles while small clusters tend to loose particles. This might seem obvious in a typical coarsening dynamics like the one in the Ising model, where smaller clusters tend to vanish and the average size of clusters increases with time. However, ordering in the original voter dynamics is quite different because same-state domains gain and loose particles at the same rate, independent on their size, and thus coarsening is only driven by fluctuations. This is due to the fact that all opinion states are equivalent in the VM on regular topologies [27] and, as consequence, the average fraction of particles in each state is conserved at all times. Therefore, the drift or net flow of particles between any two states is zero in the VM.

To study whether the transitions to a given direction are correlated with the number of particles with direction (mass ), we define the drift from direction to direction at time as

(1m)

where the sum is over all particles with direction , and is the probability that particle adopts the direction at time , calculated as the fraction of ’s neighbors with direction . Then, the net drift from small to large clusters is defined as

(1n)

where sign() is a function that takes the value () for (), and for , thus it assigns a positive weight to drifts towards larger clusters, and a negative weight to drifts towards smaller clusters. Therefore, a positive (negative) value of means that, in average, the net drift in the system is from smaller (larger) to larger (smaller) clusters. In Fig. 9(a) we show the time evolution of averaged over realizations, on a system with particles moving at speed and for various densities. We observe that is larger than zero for all times, showing that there is a net drift from small to large clusters. This generates a positive feedback in which large clusters tend to increase their size while small clusters tend to shrink, and is in contrast with the MF behavior, where no direction has a prevalence on the others and thus at all times. Therefore, the presence of a positive drift breaks the symmetry of the system and speeds up the evolution towards consensus, as compared to MF. We can check that the MF limit is achieved as increases, where we see that decreases and is already very small for .

                      
                      
Figure 9: Time evolution of four different magnitudes at zero noise (): (a) the net drift from small to large clusters, (b) the time derivative of the mean number of neighbors , (c) the ratio between the mean number of directions in MF () and in 2D () as defined in Fig. 4, and (d) the covariance between the size of a cluster and its mean degree. Simulations correspond to systems with particles, speed and densities (circles), (squares), (diamonds) and (triangles).

As we can see in Fig. 9(b), the shape of the time derivative of agrees quite well with the shape of , showing a direct relationship between clustering and drift. Also, reaches a maximum when starts to deviate from the MF curve [see Fig. 9(c)], indicating that the drift accelerates the evolution towards consensus. Figure 9 shows only the case of , but we found similar results for and as well. We speculate that the reason why larger clusters tend to increase their size more rapidly than smaller clusters is because particles that belong to larger clusters have, in average, a larger number of neighbors. And, it is known that in disordered topologies of interactions like complex networks, the weighted magnetization (, with and the state and degree of node , respectively) is conserved in the voter dynamics [27], rather than the state magnetization. This means that nodes are more likely to copy the opinion state of other nodes with large degrees, breaking the equivalence between states.

To check the relation between cluster size and mean degree we calculated the time evolution of the covariance between the mass of a given cluster and the mean degree of the particles that belong to that cluster, regardless whether the neighbors belong to the same or to a different cluster. The evolution of the covariance averaged over realizations is shown in Fig. 9(d). We observe a positive covariance between and which shows that, indeed, larger clusters have more neighbors, increasing their chances to gain more particles. A similar result was obtained in the SVM for which bigger clusters have larger mean degrees as well [28]. The fact that larger clusters tend to have a larger mean degree and grow faster, breaks the symmetry of the FVM.

6 Discussion and conclusions

We proposed and studied a flocking model where self-propelled particles interact via a simple dynamics of velocity imitation, perturbed by an external noise of amplitude . We showed that the system reaches complete order at zero noise , and that the order parameter decays continuously with in finite systems. However, in the thermodynamic limit, the precise behavior of with depends crucially on the speed of particles . At low speeds () and for a broad range of particles’ densities , the system remains completely disordered when noise is present (), even for very small amplitudes. This absence of order at finite noise is a phenomenon unseen in most related flocking models, but related to known results in the original (static) voter model. However, at high enough speeds () a completely new behavior is found. The system exhibits a second-order phase transition at a finite critical noise , from a ordered phase where for to a disordered phase where for . Simulations for a lower speed also confirm the existence of a phase transition, but at a lower critical point. However, when the speed is decreased to a value the absence of a transition is recovered. These results suggest that there is a crossover between a region for low speeds where a transition takes place at zero noise and a high-speed region where the transition happens at a noise value that increases with the speed.

It is interesting to compare this ordering behavior found in the FVM with the known behavior of the SVM. While the static version of the SVM does not display long-range order in the thermodynamic limit, the dynamic version exhibits an ordered phase at low noise for any [29]. Furthermore, the amount of order in the system for a given and seems to be quite independent of [29]. This contrasts with the results found in FVM, where order is speed dependent and only observed at high enough speeds.

We have also studied the ordering dynamics of the system at zero-noise and its approach to the fully ordered state (consensus) . We found that the dynamics is characterized by an algebraic increase with time on a first stage, and an exponential approach to the ordered state on a second stage. Interestingly, a similar ordering behavior is observed in the SVM model, although the algebraic increase happens earlier, and thus the approach to order is much faster than in the FVM. The ordering in the FVM is related to the decreasing number of different directions of motion in the system, which is well explained by the mean field theory of the multi-state voter model. The mean time to reach consensus is non-monotonic with and, as a consequence, there is an optimal density for which the system reaches full order in the shortest time. The shape of the vs curve can be qualitatively explained in terms of three main mechanisms that act on different density scales. At low densities, the dynamics is limited by the sporadic encounters between particles that introduces a delay in the interactions and leads to large consensus times. At intermediate densities, the dynamics undergoes a breaking in the symmetry of transitions between directions, induced by the spatial segregation of particles into same-direction clusters. This symmetry breaking is enhanced by the motion of particles, generating a net mass drift from small to large clusters and accelerating the approach to consensus respect to the MF behavior. At high densities the MF case of all-to-all interactions is recovered.

It has been recently shown that the flocking transition in the SVM can be understood as a liquid-gas phase transition in which the coexistence of liquid and gas is characterized by the presence of traveling density bands arranged periodically in space, and surrounded by a gas phase [23]. However, the transition found in the FVM for high speeds does not seem to be akin to the liquid-gas transition of the SVM, as we did not find any evidence of band structures in the ordered phase during the entire simulation run, i e., up to times of order . Also, bands are not observed even when the initial condition consists on all particles pointing in the same horizontal direction.

We have explored a voter-like dynamics acting on self-propelled agents subject to metric interactions, which by definition occur when two particles are less than a predefined cutoff distance apart. This type of interactions was adopted for its simplicity. Nevertheless, it is known that some social living organisms, like birds or humans, may interact in a different way, for instance by considering the first closest neighbors regardless of the distance to them [30]. It would be worthwhile to study a system with voter-like interactions in this context. In particular, the density dependence of the dynamics could turn out to be very different with respect to the one found in this article. Finally, we believe that the smooth decay of with found for the order parameter with noise calls for an explanation in terms of a simplified model. This is another topic for future investigation.

Acknowledgement

We acknowledge discussions with Dr. J. Candia, Dr. E. Albano, Dr. T. Grigera and Dr. E. Loscar. We acknowledge financial support from CONICET (PIP 11220150100039CO and PIP 0443/2014) and Agencia Nacional de Promoción Científica y Tecnológica (PICT-2015-3628) (Argentina). G. Baglietto and F. Vazquez are members of CONICET.

References

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